On connection between the switching separability of a graph and its subgraphs

A graph of order n ≥ 4 is called switching separable if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: If removing any one or two vertices of a graph always results in a switching separable subgraph then the graph itself is switching separable. On the other hand, for each odd order greater than 4, there is a graph that is not switching separable, but removing a vertex always results in a switching separable subgraph. We show some connection with similar facts on the separability of Boolean functions and the reducibility of n-ary quasigroups.